$k$-Hyperopic Cops and Robber

TL;DR

This paper introduces the $k$-hyperopic cops and robber game, a new variant where cops have limited visibility, and studies its properties, bounds, and specific cases like trees and outerplanar graphs.

Contribution

It defines the $k$-hyperopic cop number, characterizes cop-win graphs, and provides bounds and results for trees and outerplanar graphs, extending the theory of cops and robber games.

Findings

Characterization of cop-win graphs under $k$-hyperopic rules.

Upper bounds on $c_{H,k}(G)$ in terms of matching number and graph size.

$c_{H,2}(G)$ is at most 2 for outerplanar graphs.

Abstract

A generalization of hyperopic cops and robber, analogous to the $k$-visibility cops and robber, is introduced in this paper. For a positive integer $k$ the $k$-hyperopic game of cops and robber is defined similarly as the usual cops and robber game, but with the robber being omniscient and invisible to the cops that are at distance at most $k$ away from the robber. The cops win the game if, after a finite number of rounds, a cop occupies the same vertex as robber. Otherwise, robber wins. The minimum number of cops needed to win the game on a graph $G$ is the $k$-hyperopic cop number $c_{H,k}(G)$ of $G$. In addition to basic properties of the new invariant, cop-win graphs are characterized and a general upper bound in terms of the matching number of the graph is given. The invariant is also studied on trees where the upper bounds mostly depend on the relation between $k$ and the diameter of the tree. It is also proven that the 2-hyperopic cop number of outerplanar graphs is at most 2 and an upper bound in terms of the number of vertices of the graph is presented for $k \geq 3$.